in wagering, "equalizing allowance to a weaker side or player by a stronger, advantage conceded by one of the parties in proportion to the assumed chances in his favor," 1590s, found first in Shakespeare ("2 Henry IV," 1597), probably from the word's earlier sense of "condition of inequality, difference, amount by which one thing exceeds or falls short of another" (1540s), from odd (q.v.), though the exact sense evolution is uncertain. Odds was used for "unequal things, matters, or conditions" from c. 1500, and the later senses may have evolved generally from this earlier notion of "things that don't come out even."
Until 19c. treated as a singular, though obviously a plural (compare news). General sense of "chance or balance of probability in favor of something happening" is by 1580s. Sense of "disagreement, variance, strife" (1580s) is the notion in at odds "at controversy or quarrel, unable to agree." Odds-on "on which the odds are laid" is by 1890.